= Tau-pair missing-mass teaching model
Choose $m_\tau=2\,\mathrm{GeV}$ and $\sqrt{s}=6\,\mathrm{GeV}$ purely for illustration. Then each parent has $E_\tau=3\,\mathrm{GeV}$, $|\mathbf p_\tau|=\sqrt5\,\mathrm{GeV}$ and $\beta=\sqrt5/3$. Treat all six daughters as massless, including the charged <Electron> and <antimuon>, and generate each three-body <particle decay> with constant squared <decay amplitude>. Choose the parent direction isotropically and the two decays independently; this omits the physical production angular law and <spin> correlations.
The <Lorentz-invariant phase space> factorizes through $t=q^2$ into a two-body decay to $\ell+q$, followed by a two-body decay of $q$:
$$
d\Phi_3=\frac{dt}{2\pi}\,d\Phi_2(P_\tau;p_\ell,q)\,d\Phi_2(q;q_1,q_2).
$$
After angular integration the first factor contributes $1-t/m_\tau^2=x$, while the massless second factor is constant. Since $|dt|=m_\tau^2dx$, the normalized chosen density is $f(x)=2x$. Drawing $x=\sqrt U$ with $U$ uniform on $[0,1]$ implements this density. Isotropic directions, followed by a <Lorentz boost> for the invisible pair and another for the parent, conserve the complete <four-momentum> of each generated event.
\Image[/tau-pair-missing-mass-phase-space.png]
{title=Three-body phase space spreads visible energies and invisible recoil}
{description=Original fixed-seed teaching sample with chosen masses, massless daughters and a constant decay amplitude. The two-body control has all outgoing four-momentum visible. The transverse acoplanarity convention is $180^\circ-\Delta\phi$; this is a geometric diagnostic, without the historical selection cuts. These are not historical data or a true weak-decay spectrum.}
Define transverse acoplanarity by $\alpha=\pi-\Delta\phi$, where $\Delta\phi\in[0,\pi]$ is the angle between the two visible <momentum> projections perpendicular to the beam. A fully visible two-body event with zero initial transverse <momentum> has $\alpha=0$ and $M_{\rm miss}^2=0$. The toy <tau lepton> events spread away from that point. These observables are complementary: an angular imbalance asks about directions, while the <missing invariant mass> also uses the visible <energies> and longitudinal <momenta>.
A <Lorentz-invariant phase space> density supplies allowed configurations and their geometric measure. A physical <weak interaction> <decay amplitude> reweights them and correlates <spins>. Consequently the plotted density is a lesson about conservation and available configurations, not a predicted experimental rate or a fit to the discovery sample. The massless charged-daughter approximation is also unsuitable near the <muon> rest-energy threshold.
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